OCR Physics A
25.2 Nuclear decay equations
Application
Modelling radioactive decay
Specification references
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6.4.3 g)
HSW3
M0.5, M2.5, M3.9
Introduction
Many of the chemical elements present in the Earth’s crust and atmosphere are
radioactive – their nuclei are unstable, resulting in the emission of alpha, beta, or
gamma radiation until they eventually become stable. For some decay processes,
this process happens quickly, with the nuclei changing from one type to another in
tiny fractions of a second. In other cases, this process may take many billions of
years, with the process of radioactive decay being incredibly slow. The time taken
for the activity or number of radioactive nuclei to reduce to 50% of the original value
is called its half-life and this is related to the decay constant of the nucleus in
question. In this activity you will consider the relationship between the half-life of a
source and its decay constant, as well as perform a task to model the behaviour of a
radioactive source using data in a spreadsheet.
Learning outcome
After completing this worksheet you will be able to:

use graphical methods and spreadsheets to model the equation
ΔN
Δt
  N for
the process of radioactive decay.
Background
The equation that governs the number of radioactive nuclei left after time t for a radioactive
isotope is given by N  Noet, where N is the number of nuclei that are still radioactive after
time t, No is the original number of radioactive nuclei present at time t  0, t is the time of
decay, and λ is the decay constant of the radioisotope in question. The significance of the
decay constant, , is that it relates to the probability that a single radioactive nucleus will
undergo a decay by emitting alpha, beta, or gamma radiation. The greater the value of the
decay constant, the more likely it is to emit radiation, so the more radioactive the source will
be and the more unstable the nucleus can be said to be. Some very unstable elements, such
as californium, have decay constants that are extremely high, meaning that they decay into
other elements very quickly – within fractions of billionths of a second. Conversely, some
radioactive nuclei are much more stable, with decay constants that are very small.
© Oxford University Press 2016 http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
1
25.2 Nuclear decay equations
Application
OCR Physics A
There is a simple mathematical relationship between the decay constant of a
radionuclide and its half-life. The relationship is one of inverse proportion – as the
decay constant increases, the half-life of the source decreases. Technetium-99m
has a half-life of 6 hours and a relatively large decay constant; carbon-14 has a halflife of 5760 years, and a decay constant that is significantly lower.
Task
As with the discharging of a charged capacitor, the decay of a radioactive source
also follows a negative exponential equation. This radioactive decay can be
modelled using an iterative (step-by-step) process. The equation that can be used in
this process, and the filling in of the associated spreadsheet, is given by
ΔN
Δt
= - N
In order to model the decay, and fill in the table, the following steps need to be
taken:
Step 1: Write the initial number of undecayed nuclei in the second column.
ΔN
by calculating N, which is the decay constant multiplied by the
Δt
number of particles present at time t. In this example,   0.05 s1.
Step 2: Find
Step 3: Find ΔN by multiplying
ΔN
Δt
by Δt , which in this case is 0.5 s.
Step 4: Find the new value for N by adding the previous value of N to ΔN.
Step 5: Move this new value of N to the next row and repeat the process.
Question: How many radioactive nuclei will be left after t  2.0 seconds if there
are 4.80 × 1024 radioactive nuclei present at t  0, given that the decay constant
  0.05 s−1?
Time/s
No. of undecayed
nuclei
ΔN
Δt
N
N
0.0
4.80 × 1024
2.40 × 1023
1.20 × 1023
4.68 × 1024
0.5
4.68 × 1024
1.0
1.5
2.0
© Oxford University Press 2016 http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
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25.2 Nuclear decay equations
Application
OCR Physics A
It is also possible to obtain the value for the decay constant by plotting a graph of
ln N versus time. You can obtain values for the decay constant from the gradient of
a ln N versus time graph or from a ln A versus time graph, where A is the activity of
the source. Complete the table below (the first values have been added for you) and
plot suitable graphs.
Use your graph to find the value of .
NB, the activity A is the value of
ΔN
(ignore the minus sign in front of the value
Δt
when finding the natural logarithm, as it is not possible to obtain a value for a
logarithm of a negative number).
Time/s
0.0
ln N
56.83
ln A
53.835
0.5
1.0
1.5
2.0
(12 marks)
© Oxford University Press 2016 http://www.oxfordsecondary.co.uk/acknowledgements
This resource sheet may have been changed from the original
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